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   "source": [
    "# 2D Linear Advection with Finite Differences\n",
    "## CH EN 6355 - Computational Fluid Dynamics\n",
    "**Prof. Tony Saad (<a>www.tsaad.net</a>) <br/>Department of Chemical Engineering <br/>University of Utah**\n",
    "<hr/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format = 'svg'\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.animation as animation\n",
    "plt.rcParams['animation.html'] = 'html5'\n",
    "from matplotlib import cm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "nx = 21\n",
    "ny = 21\n",
    "Lx = 1.0\n",
    "Ly = 1.0\n",
    "x = np.linspace(0,Lx,nx)\n",
    "y = np.linspace(0,Ly,ny)\n",
    "dx = Lx/(nx-1)\n",
    "dy = Ly/(ny-1)\n",
    "# create a grid of coordinates\n",
    "xx,yy = np.meshgrid(x,y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
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       "  <clipPath id=\"p94437bc12a\">\n",
       "   <rect height=\"217.44\" width=\"334.8\" x=\"30.103125\" y=\"10.999219\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ω0 = 2.0*np.pi\n",
    "u0 = lambda x,y : np.sin(ω0*x) * np.sin(ω0*y)\n",
    "# plot the initial condition\n",
    "plt.contourf(xx,yy,u0(xx,yy))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "cx = 1.0\n",
    "cy = 1.0\n",
    "\n",
    "dt = 0.001\n",
    "tend = 0.5 #s\n",
    "t = 0\n",
    "\n",
    "cflx = cx * dt/dx\n",
    "cfly = cy * dt/dy\n",
    "\n",
    "# setup the initial condition\n",
    "sol = []\n",
    "u = np.zeros([ny+2,nx+2]) # we will ghost cells to simplify the implementation of periodic BCs\n",
    "u[1:-1, 1:-1] = u0(xx,yy)\n",
    "# set periodic boundaries\n",
    "u[:,0] = u[:,-3] #x-minus face\n",
    "u[:,-1] = u[:,2] #x-plus face\n",
    "u[0,:] = u[-3,:] #y-minus face\n",
    "u[-1,:] = u[2,:] #y-plus face\n",
    "sol.append(u)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "while t < tend:\n",
    "    un = sol[-1]\n",
    "    unew = un.copy()\n",
    "    unew[1:-1,1:-1] = un[1:-1,1:-1] - cflx * (un[1:-1,1:-1] - un[1:-1,:-2]) - cfly * (un[1:-1,1:-1] - un[:-2,1:-1])\n",
    "    # set periodic boundaries\n",
    "    unew[:,0] = unew[:,-3] #x-minus face\n",
    "    unew[:,-1] = unew[:,2] #x-plus face\n",
    "    unew[0,:] = unew[-3,:] #y-minus face\n",
    "    unew[-1,:] = unew[2,:] #y-plus face\n",
    "\n",
    "    sol.append(unew)\n",
    "    t += dt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
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    "fig = plt.figure(figsize=(6.1,5),facecolor='w')\n",
    "ims = []\n",
    "levs = np.linspace(-1,1,20)\n",
    "i = 0\n",
    "t = 0.0\n",
    "for solution in sol:    \n",
    "    if (i%10==0):        \n",
    "        im = plt.contourf(xx,yy,solution[1:-1,1:-1],cmap=cm.jet,levels=levs,vmax=1.0,vmin=-1.0)\n",
    "        ims.append(im.collections)\n",
    "    i+=1\n",
    "    t += dt\n",
    "    \n",
    "cbar = plt.colorbar()\n",
    "plt.clim(-1,1)\n",
    "cbar.set_ticks(np.linspace(-1,1,5))\n",
    "\n",
    "ani = animation.ArtistAnimation(fig, ims, interval=35, blit=True,repeat_delay=1000)\n",
    "\n",
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